Properties

Label 2.2.8.1-1458.1-x1
Base field \(\Q(\sqrt{2}) \)
Conductor norm \( 1458 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))
 
gp: K = nfinit(Polrev([-2, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(3a-3\right){x}+1\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([-3,3]),K([1,0])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([0,0]),Polrev([-3,3]),Polrev([1,0])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![-3,3],K![1,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((27a)\) = \((a)\cdot(3)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1458 \) = \(2\cdot9^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3456a)\) = \((a)^{15}\cdot(3)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -23887872 \) = \(-2^{15}\cdot9^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{211977}{256} a + \frac{46143}{64} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(2 a + 2 : -6 a - 11 : 1\right)$
Height \(0.19839328773173120482735367987692173374\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-2 a + 2 : 2 a - 3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.19839328773173120482735367987692173374 \)
Period: \( 13.455317071958475846751479410880232406 \)
Tamagawa product: \( 15 \)  =  \(( 3 \cdot 5 )\cdot1\)
Torsion order: \(3\)
Leading coefficient: \( 3.1459706209427574486516159027663782385 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((a)\) \(2\) \(15\) \(I_{15}\) Split multiplicative \(-1\) \(1\) \(15\) \(15\)
\((3)\) \(9\) \(1\) \(II\) Additive \(1\) \(3\) \(3\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 1458.1-x consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.